L ECTURE 11: D ISCRETE -T IME D YNAMICAL S YSTEMS 2 I NSTRUCTOR : G - - PowerPoint PPT Presentation

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LECTURE 11: DISCRETE-TIME DYNAMICAL SYSTEMS 2

INSTRUCTOR: GIANNI A. DI CARO

15-382 COLLECTIVE INTELLIGENCE – S18

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OCCURRENCE OF PERIODIC WINDOWS FOR 𝑠 > 𝑠

#

§ At 𝑠 ≈ 𝑠

# = 3.57 the map becomes

chaotic and the attractor changes from a finite to an infinite set of points § The large window at 𝑠 ≈ 3.83 contains a stable period-3 orbit

§ 𝑔 𝑦 = 𝑠𝑦 1 − 𝑦 à the logistic map is 𝑦012 = 𝑔(𝑦0) § 𝑦015 = 𝑔 𝑔 𝑦0 , 𝑦017= 𝑔(𝑔 𝑔 𝑦0 ) = 𝑔7(𝑦0) § We are looking for 3-period cycles: every point 𝑞 in a 3-period cycle repeats every 3 iterates § à 𝑞 must satisfy 𝑞 = 𝑔7(𝑞) à 𝑞 is a fixed-point of the 𝑔7 map § Unfortunately, the 𝑔7 map is an 8-degree polynomial, a bit complex to study

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OCCURRENCE OF PERIODIC WINDOWS FOR 𝑠 > 𝑠

#

𝑠 = 3.835, inside 3-period window

§ Intersections between the graph and the diagonal correspond to the solutions of 𝑔7 𝑦 = 𝑦 § Only the black dots correspond to fixed points, and there are 3 of them, corresponding to the the 3-period cycle § The slope of the function, |𝑔:| is greater than 1 for the white dots, and less than 1 for the black ones § For the other intersections, they correspond to fixed points or 1-period

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OCCURRENCE OF PERIODIC WINDOWS FOR 𝑠 > 𝑠

#

§ The 6 intersections of interest have vanished! § Not anymore periodic behavior § For some 𝑠 between 3.8 and 3.835 the graph is tangent to the diagonal § At this critical value of 𝑠, the stable and unstable 3-period cycles coalesce and annihilate in a tangent bifurcation, that sets the beginning of the periodic window § It can be computed analytically that this happens at 𝑠 = 1 + 6

𝑠 = 3.835, inside 3-period window 𝑠 = 3.8, before 3-period window

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OCCURRENCE OF PERIODIC WINDOWS FOR 𝑠 > 𝑠

#

§ Just after the tangent bifurcation, the slope at black dots (periodic points) is ≈ +1 (a bit less) § For increasing values of 𝑠, hills and valleys become steeper / deeper § The slope of 𝑔7 at the black dots decreases steadily from ≈ +1 to -1. When this occurs, a flip bifurcation happens, that causes each of the fixed periodic points to split in two § à the 3-period cycle becomes a 6-period cycle! § … the process iterates, bringing the period doubling cascade!

𝑠 = 3.835, inside 3-period window 𝑠 > 3.835

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WHAT ABOUT SENSITIVITY TO INITIAL CONDITIONS?

Chaos requires aperiodic orbits + sensitivity to initial conditions ü Aperiodic orbits arise § Sensitivity to initial conditions? § Given an initial condition 𝑦=, and an nearby point 𝑦=+𝜀=, 𝜀= ≈ 0 § 𝜀 expresses the separation between two near initial conditions § 𝜀0 is the separation after 𝑜 iterates § If |𝜀0| ≈ |𝜀=|𝑓0B then 𝜇 is called the Lyapounov exponent (of 𝑦=) § A positive Lyapounov exponent is a signature of chaos § Applies also to flows (like in the figure)

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LYAPOUNOV EXPONENTS

§ |𝜀0| ≈ |𝜀=|𝑓0B, taking the logarithm § 𝜇 ≈

2 0 ln| FG FH |

§ We can observe that 𝜀0 = 𝑔0 𝑦= + 𝜀= − 𝑔0(𝑦=) § 𝜇 ≈

2 0 ln| IG JH1FH KIG(JH) FH

| =

2 0 ln |(𝑔0): 𝑦= |, for 𝜀= → 0

§ 𝑔0 is a function of all 𝑦M, 𝑗 = 0, …,𝑜 à expansion by chain rule § 𝜇 ≈

2 0 ln| ∏

(𝑔0): 𝑦M

0K2 MQ=

| =

2 0 ∑0K2 MQ=

ln | (𝑔0): 𝑦M | § If the limit for 𝑜 → ∞ exists, this is the Lyapounov exponent for the orbit starting in 𝑦= § 𝜇 is the same for all points in the basin of attraction of an attractor § For stable fixed points and cycles, 𝝁 < 𝟏 § For chaotic attractors, 𝝁 > 𝟏

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LYPUNOUV EXPONENTS FOR THE LOGISTIC MAP

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ANOTHER MAP: HENON MAP

Classical settings for getting a chaotic behavior: 𝑏 = 1.4, 𝑐 = 0.3 Fractal dimension ≅1.26

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FROM N-DIMENSIONAL MAPS TO CELLULAR AUTOMATA

§ Let’s introduce topological notions and constraints § Each variable represents the time-evolution

• f a spatial location

§ Two variables can be (or not) neighbors § Neighboring concepts go beyond metric spaces § A variable’s evolution only depends on its neighbors à Influential variables § Not everybody is neighbor of everybody § à Each map only depends on a restricted set of neighbors, but the entire systems stays coupled Cellular Automata (CA) 𝑦012

2

= 𝑔

2(𝑦0 2, 𝑦0 5, … .𝑦0 [)

𝑦012

5

= 𝑔

5(𝑦0 2, 𝑦0 5, …. 𝑦0 [)

𝑦012

[

= 𝑔

[(𝑦0 2, 𝑦0 5, … .𝑦0 [)

….. § Mathematical simplification § Modeling spatial relations § Dynamical model of many real-world systems

§ Generic k-dimensional map …maybe too generic (complex!)

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STATE VARIABLES SPATIALLY BOUNDED ON LATTICES: CELLULAR AUTOMATA

§ State variable ↔ State of Spatial location / Cell § Cell in a 1D, or 2D, or 3D Lattice 1D 2D

𝑦2 𝑦5 𝑦7 𝑦M 𝑦MK2 𝑦M12 𝑦[ Example of selected neighborhood of 𝑦M, represented by the set {𝑦MK2, 𝑦M12} 𝑦2 𝑦5 𝑦7 𝑦[ 𝑦[12𝑦[15 𝑦[17 𝑦[1] Example of selected neighborhood of 𝑦[1M, represented by the set {𝑦[1MK2, 𝑦[1M12, 𝑦M, 𝑦M12,𝑦MK2, 𝑦5[1M, 𝑦5[1M12, 𝑦5[1MK2}

𝑦[1M

𝑦5[

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CAS ARE LATTICE MODELS

§ Regular 𝒐-dimensional discretization of a continuum § E.g., an 𝑜-dimensional grid § Periodic (toroidal) or non periodic structure § More abstract definition: Regular tiling of a space by a primitive cell

3D grid lattice, filled with spheres

• f different colors

Bethe lattice, ∞-connected cycle-free graph where each node is connected to 𝑨 neighbours, where 𝑨 is called the coordination number

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CELLULAR AUTOMATA

1D

𝑦2 𝑦5 𝑦7 𝑦M 𝑦MK2 𝑦M12 𝑦[ Example of selected neighborhood of 𝑦M, represented by the set {𝑦MK2, 𝑦M12}

𝑦012

2

= 𝑔

2(𝑦0 2, 𝑦0 5, … .𝑦0 [)

𝑦012

5

= 𝑔

5(𝑦0 2, 𝑦0 5, …. 𝑦0 [)

𝑦012

[

= 𝑔

[(𝑦0 2, 𝑦0 5, … .𝑦0 [)

….. 𝑦012

2

= 𝑔

2(𝑦0 2, 𝑦0 5)

𝑦012

5

= 𝑔

5(𝑦0 2, 𝑦0 5, 𝑦0 7)

𝑦012

M

= 𝑔

M(𝑦0 MK2, 𝑦0 M , 𝑦0 M12)

….. 𝑦012

[

= 𝑔

[(𝑦0 [K2, 𝑦0 [)

….. 𝑦012

2

= 𝑔

2(𝑦0 [, 𝑦0 2, 𝑦0 5)

𝑦012

M

= 𝑔

M(𝑦0 MK2, 𝑦0 M , 𝑦0 M12)

….. 𝑦012

[

= 𝑔

[(𝑦0 [K2, 𝑦0 [, 𝑦0 2)

….. Toroidal boundary conditions 𝑦012

2

= 𝑔(𝑦0

[, 𝑦0 2, 𝑦0 5)

𝑦012

M

= 𝑔(𝑦0

MK2, 𝑦0 M , 𝑦0 M12)

….. 𝑦012

[

= 𝑔(𝑦0

[K2, 𝑦0 [, 𝑦0 2)

Single map ….. …..

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CAS HISTORICAL NOTES